Estimations of the Best Approximations for the Classes of Infinitely Differentiable Functions in Uniform and Integral Metrics

Abstract

We establish uniform (with respect to the parameter p, 1 ≤ p ≤ ∞) upper estimations of the best approximations by trigonometric polynomials for the classes C _β,p ^ψ of periodic functions generated by sequences ψ(k) vanishing faster than any power function. The obtained estimations are exact in order and contain constants expressed in the explicit form and depending solely on the function ψ. Similar estimations are obtained for the best approximations of the classes L _β,1 ^ψ in metrics of the spaces L _s , 1 ≤ s ≤ ∞.